How do I know that the DQN has learnt an appropriate Q function?
DQN is famous for doing over-approximation on Q function. However, having over approximated Q does not imply that it does not perform well in the environment. (unless it looks ridiculously high) From my experience, high learning rate usually cause over approximated Q, or mistakes made in the code. Best way to check is to see plot of Q function when running on environment to see if it makes any sense. (e.g. It should go lower for bad states and vice versa)